Problem: $ F = \left[\begin{array}{rr}1 & -1 \\ 3 & 2\end{array}\right]$ $ E = \left[\begin{array}{rrr}-2 & 4 & 0 \\ 2 & -1 & 3\end{array}\right]$ What is $ F E$ ?
Solution: Because $ F$ has dimensions $(2\times2)$ and $ E$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ F E = \left[\begin{array}{rr}{1} & {-1} \\ {3} & {2}\end{array}\right] \left[\begin{array}{rrr}{-2} & \color{#DF0030}{4} & \color{#9D38BD}{0} \\ {2} & \color{#DF0030}{-1} & \color{#9D38BD}{3}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-2}+{-1}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-2}+{-1}\cdot{2} & ? & ? \\ {3}\cdot{-2}+{2}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-2}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{4}+{-1}\cdot\color{#DF0030}{-1} & ? \\ {3}\cdot{-2}+{2}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{-2}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{4}+{-1}\cdot\color{#DF0030}{-1} & {1}\cdot\color{#9D38BD}{0}+{-1}\cdot\color{#9D38BD}{3} \\ {3}\cdot{-2}+{2}\cdot{2} & {3}\cdot\color{#DF0030}{4}+{2}\cdot\color{#DF0030}{-1} & {3}\cdot\color{#9D38BD}{0}+{2}\cdot\color{#9D38BD}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-4 & 5 & -3 \\ -2 & 10 & 6\end{array}\right] $